Triangle calculator SAS

Acute scalene triangle.

Sides: a = 90 b = 100 c = 95.39439201417

Area: T = 3897.114431703
Perimeter: p = 285.3943920142
Semiperimeter: s = 142.6976960071

Angle ∠ A = α = 54.79112808971° = 54°47'29″ = 0.9566288253 rad
Angle ∠ B = β = 65.20987191029° = 65°12'31″ = 1.13881068494 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: h_a = 86.60325403784
Height: h_b = 77.94222863406
Height: h_c = 81.70657169103

Median: m_a = 86.74767578645
Median: m_b = 78.10224967591
Median: m_c = 82.31103881658

Inradius: r = 27.31104228366
Circumradius: R = 55.07657054729

Vertex coordinates: A[95.39439201417; 0] B[0; 0] C[37.7388254122; 81.70657169103]
Centroid: CG[44.37773914212; 27.23552389701]
Coordinates of the circumscribed circle: U[47.69769600708; 27.53878527364]
Coordinates of the inscribed circle: I[42.69769600708; 27.31104228366]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.2098719103° = 125°12'31″ = 0.9566288253 rad
∠ B' = β' = 114.7911280897° = 114°47'29″ = 1.13881068494 rad
∠ C' = γ' = 120° = 1.04771975512 rad

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How did we calculate this triangle?

1. Calculation of the third side c of the triangle using a Law of Cosines

$a = 90 ; ; b = 100 ; ; gamma = 60° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 90**2+100**2 - 2 * 90 * 100 * cos 60° } ; ; c = 95.39 ; ;$

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

$a = 90 ; ; b = 100 ; ; c = 95.39 ; ;$

2. The triangle circumference is the sum of the lengths of its three sides

$p = a+b+c = 90+100+95.39 = 285.39 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 285.39 }{ 2 } = 142.7 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 142.7 * (142.7-90)(142.7-100)(142.7-95.39) } ; ; T = sqrt{ 15187500 } = 3897.11 ; ;$

5. Calculate the heights of the triangle from its area.

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3897.11 }{ 90 } = 86.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3897.11 }{ 100 } = 77.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3897.11 }{ 95.39 } = 81.71 ; ;$

6. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 100**2+95.39**2-90**2 }{ 2 * 100 * 95.39 } ) = 54° 47'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 90**2+95.39**2-100**2 }{ 2 * 90 * 95.39 } ) = 65° 12'31" ; ; gamma = 180° - alpha - beta = 180° - 54° 47'29" - 65° 12'31" = 60° ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 3897.11 }{ 142.7 } = 27.31 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 90 }{ 2 * sin 54° 47'29" } = 55.08 ; ;$

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 95.39**2 - 90**2 } }{ 2 } = 86.747 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 95.39**2+2 * 90**2 - 100**2 } }{ 2 } = 78.102 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 90**2 - 95.39**2 } }{ 2 } = 82.31 ; ;$
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